About asymptotics and large deviations in the central limit theorem for sums $\sum \mathbf{f(q^nt)}$

In the paper new estimates for the remainder term and large deviations in the central limit theorem for sums $\sum \mathbf{f(q^nt)}$ are obtained. Technique and methods of reasoning contained not only in the classical probability theory, but also in the theory of diophantine equations with exponential function are used in the given proofs.